In Blender 2.7 or 2.8 how can you slice off parts of a 2D grease pencil drawing and move it to its own layer?
Begin editing the GP strokes. In 2.80 you select the GP objects and go into edit mode, while in 2.79 you need to select Edit Strokes in the mode menu.
Select the curve points that you want to move to another layer.
Press M and either choose an existing layer or New Layer.